Numerical approximation of fractional diffusion equations on metric graphs

2026/08/03 by David Bolin, Lenin Riera-Segura, Alexandre B. Simas
Mathematics · Computer Science · #math.NA #cs.NA #msc:65J08 #msc:65M60 #msc:65M15 #msc:65M12 #msc:35R02 #msc:35R11

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arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

We study fractional diffusion equations on compact metric graphs, where the nonlocal dynamics is governed by fractional powers of the shifted Kirchhoff-Laplacian. Building on recent advances in the analysis of fractional operators on metric graphs, we establish a rigorous mathematical framework and propose a fully discrete scheme based on backward Euler time-stepping and finite element discretization. To approximate the action of the fractional operator, we employ rational approximations, reducing the problem to a sequence of sparse elliptic solves for efficient implementation. We derive error estimates for the temporal, spatial, and rational discretizations, and confirm convergence through numerical experiments.

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